Outer measure ( for better understanding) скачать в хорошем качестве

Outer measure ( for better understanding)

Let I be a nonempty interval of real numbers. We define length of I by L(I) where I is open and bounded interval and otherwise define its length to be the difference of its endpoints. For a set A of real numbers, consider the countable collections of I k nonempty open, bounded intervals that cover A, that is, collections for which A is contained each such collection, consider the sum of the lengths of the intervals in the collection. Since the lengths are positive numbers, each sum is uniquely defined independently of the order of the terms. We define the outer measure of A, m*( A), to be the infimum of all such sums. #outermeasure #skclasses #puneuniversity #lebesguemeasure #measuretheory #realanalysis #outermeasureexample #hindi #statistics #measuretheory_statistics What is outer measure? Outer measure in Hindi Let I be a nonempty interval of real numbers. We define its length, 1(I), to be oo if I is unbounded and otherwise define its length to be the difference of its endpoints. For a set A of real numbers, consider the countable collections {Ik}'1 of nonempty open, bounded intervals that cover A, that is, collections for which A C U' 1 Ik. For each such collection, consider the sum of the lengths of the intervals in the collection. Since the lengths are positive numbers, each sum is uniquely defined independently of the order of the terms. We define the outer measure3 of A, m* (A), to be the infimum of all such sums, that is m*(A) = inf I(Ik) 1k=1 ACCUHIk . It follows immediately from the definition of outer measure that m* (0) = 0. Moreover, since any cover of a set B is also a cover of any subset of B, outer measure is monotone in the sense that ifACB, then m*(A) m*(B). In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical condition